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A proof that special unitary groups over a field $F$ are generated by transvections -- except when $n = 3$ and $\# F = 4$ in which case the result is not true -- is given in Larry Grove's text: see Theorem 11.15 on page 104.

The simplicity of the projective special unitary group $PSU(V)$ -- except when $(\operatorname{dim} V, \# F)$ is one of $(2,4)$, $(2,9)$ and $(3,4)$ in which cases the result is not true -- is Theorem 11.26 on page 108 of Grove's text.